MBA Math Review

1. MBA Math Review Module 2 Derivatives Optimization Functions of Several Variables

2. t = 1 t = 0 t = 2 t = 3 A Simple Example • Suppose we observe an apple falling from a (very tall) tree. • Physics says after t seconds the apple falls 16t2 feet. • How fast is the apple falling after t seconds?

3. Average Velocity • AV = (total distance traveled)/(time taken) • What is the average velocity from • t = 1 to t = 1.5 • t = 1 to t = 1.1 • t = 1 to t = 1.01 • t = 1 to t = 1.001 • What do you think the instantaneous velocity should be at t=1?

4. What is a derivative? Slope of secant line is average rate of change between x and x+h. f(x+h) f(x) x x+h

5. Let h get very small. The secant line gets very close to the a tangent line. The average rate of change becomes an instantaneous rate of change. f(x+h) f(x) x x+h

6. Let h get very small. The secant line gets very close to the a tangent line. The average rate of change becomes an instantaneous rate of change. f(x+h) f(x) x x+h

7. Let h get very small. The secant line gets very close to the a tangent line. The average rate of change becomes an instantaneous rate of change. f(x+h) f(x) x x+h

8. Presto! The Derivative! • Slope of tangent line is instantaneous rate of change. Slope = f’(x) f(x) x

9. What is a Derivative? • Two Interpretations: • The slope of a tangent line to the curve. • The rate at which the function is changing. • Notice: • How fast the curve changes depends on x. • The derivative is a function of x.

10. Ways of Writing Derivatives “f prime of x.” This notation is popular because it is compact. “df, dx” This notation highlights the derivative as the “rate of change” of f. “partial f, partial x” This means the same thing as df/dx. It is usually used when f depends on other variables in addition to x.

11. Example 10.0 • f(x)=x2 • f’(x) = 2x • The slope of the tangent line depends on where it is drawn. 7.5 5.0 2.5 0.0 0 1 2 3 x

12. Why Derivatives are Useful • The rate of change is a useful thing to know. • The derivative of cost is Marginal Cost • They tell when a function is increasing or decreasing. • f’(x) > 0: f(x) is increasing • f’(x) < 0: f(x) is decreasing • Most useful of all: Optimization!

13. Some Rules for Derivatives • Most basic rule: axn = naxn-1 • Example: f(x) = 3x2, f’(x) = 6x • Write radicals/rational functions using fractional/negative exponents: • f(x) = 1/x = x-1 f’(x) = -x-2 = -1/x2

14. Sums and Differences • General Rule: • f(x) = g(x) + h(x) f’(x) = g’(x) + h’(x) • Examples: • f(x) = 2x2 + 3x f’(x) = 4x + 3

15. Chain Rule • Chain rule: • f(x) = g(h(x)) • f’(x) = g’(h(x)) h’(x) • Example: Derivative of thing inside sqrt. Derivative of sqrt.

16. Take the derivative of the following: Note the derivative of ln(x) is 1/x. Examples

17. What to Know • Know what derivatives mean. • Know how to take the derivative of a polynomial. • Know the chain rule. • Look up everything else.

18. 11.5 9.0 • If f’(x)>0 then f(x) is increasing move left. 6.5 • If f’(x)=0 then f(x) is neither increasing nor decreasing you’re there! 4.0 1.5 -1.0 -4 -3 -2 -1 0 1 2 3 4 x Optimization • To get to a minimum: • If f’(x) <0 then f(x) is decreasing move to the right.

19. Example: Maximize Profit • Manufacturing x TV sets per month costs • C(x) = 72,000 + 60x • The number of TV sets demanded each month depends on price. The relationship is • for( 0 < x < 6000) • How many TV sets should you produce? • How should you price them?

20. Example: Microeconomics • Marginal Cost: • Extra cost required to produce one more item. • Marginal Revenue: • Extra revenue from producing one more item. • General Profit Maximization Rule: • Marginal Cost = Marginal Revenue • Why?

21. Economics: Marginal Cost • Let C(x) be the total cost of producing x units. • The marginal cost is C’(x). • Marginal cost approximates the cost of producing one more unit.

22. Example: Marginal Cost • Total cost of producing x sailboats: C(x) = 500 + 24x - 0.2x2 • Find a formula for marginal cost. • What is marginal cost if you currently produce 30 sailboats. • What is the exact cost of producing the 31st sailboat?

23. but it’s a pretty good approximation. 1060 1055 1050 1045 1040 30 30.5 31 31.5 X Marginal Cost: The Picture Marginal cost is only an approximation to the cost of the next unit… 1300 1200 1100 1000 900 800 700 600 500 0 10 20 30 40 50 X

24. Marginal Revenue = Marg. Cost • Let x represent the amount of output you produce and notice profit = revenue - cost. • P(x) = R(x) - C(x). • To find optimal x, take derivative of profit and set to 0. • P’(x) = R’(x) - C’(x) = 0 • R’(x) = C’(x) • Marginal Revenue = Marginal Cost

25. Example: Vegas Baby!! • A 200 room hotel in Vegas is filled to capacity at $40 per room. • For each $1 increase in rent, 4 fewer rooms are rented. • Each rented room costs $8/day to service. • What rent should they charge? • What profit should they expect?

26. Partial Derivatives • In real life functions have several inputs. • The amount you can produce depends on capital, labor, and materials. • You can’t graph a function of several variables. • Imagine changing one variable while holding the others constant.

27. Partial Derivative Example • You fit a statistical model to the following • FC: car’s fuel consumption in gal/1000 mi • HP: car’s horsepower • W: car’s weight in pounds • IN: car’s length in inches. • The model: FC = -11.5 + 0.078HP + 0.021W + 0.10IN -0.000059INW

28. Question • What impact do you expect increasing weight by 1 lb. will have on fuel consumption, holding all else constant? • Take derivative with respect to W, treating other variables as constants. FC = -11.5 + 0.078HP + 0.021W + 0.10IN -0.000059INW FC’ = 0.021-0.000059IN